Objectives

• Introduce Bayesian Linear Regression (BLR): Understand its principles and how it differs from traditional methods.

• Explain Bayesian Concepts: Highlight Bayes’ Theorem, prior knowledge, and posterior distributions.

• Discuss Practical Applications: Show how BLR is applied in analyzing real-world data, like airline delays.

• Explore Advantages of Bayesian Methods: Quantifying uncertainty, improving predictions, and handling complex data.

• Present Analysis Findings: Summarize key insights from our BLR model on weather-related airline delays.

What is Bayesian Linear Regression?

• BLR: A statistical approach combining prior knowledge and new data.

• Goal: Model relationships, make predictions, and handle uncertainty in estimates.

• Difference from Traditional Methods: Probability-based estimates instead of fixed values.

Introduction to Bayesian Linear Regression

• Regression under the frequentist framework
  • Independent variables are used to predict dependent variables
  • Linear regression finds best-fitting line to observed data to make further predictions
    • Regression parameters (\beta) are assumed to be fixed
  • Only collected data is used for approximation
• Regression under the Bayesian framework
  • Independent variables are used to predict dependent variables
  • Regression parameters (\beta) are not assumed to be fixed
  • Collected data is used alongside prior knowledge for approximation

Why Bayesian?

Advantages of Bayesian Linear Regression[1]

• Incorporation of Prior Knowledge

• Uncertainty Quantification

• Expanded Hypotheses

• Automatic Meta-Analyses

• Improved Handling of Small Samples

• Complex Model Estimation

Steps in Bayesian Linear Regression

1. Model Specification: Define the linear relationship between the dependent and independent variables.

2. Choose Priors: Select prior distributions for the model parameters, reflecting any existing knowledge about their values.

3. Data Collection: Gather relevant data for the variables in the model.

4. Model Fitting: Use computational methods, such as Markov Chain Monte Carlo (MCMC), to estimate the posterior distributions of the parameters based on the observed data.

5. Result Interpretation: Analyze the posterior distributions to understand the relationships between variables, including estimating means and credible intervals.

Heather’s Prior Selection & Model Specification

Prior Selection

• Intercept (\beta_0): \beta_0 \sim N(0, 5^2) Assumes no strong baseline effect.

• Slope (\beta_1): \beta_1 \sim N(0, 5^2) Reflects no strong prior belief about the relationship between weather incidents and delays.

• Error Term (\sigma): \sigma \sim \text{Exp}(1) Accounts for variability in delays; allows flexibility.

Model Specification

Y_i \mid \beta_0, \beta_1, \sigma \sim N(\mu_i, \sigma^2) \mu_i = \beta_0 + \beta_1 X_i

• Y_i: Arrival delay (minutes)
  
• X_i: Weather-related incidents

Meet My Dataset!

Exploring the Data

Exploring the Data

Choosing Focus

Table 1: Summary of Flight Arrivals, Delays, Cancellations, and Diversions

Code for Model

Trace Plots and Posterior Distributions

Model Parameters and Estimates

Model Diagnostics and Fit Statistics

Posterior Distribution for Weather Count Coefficient

Posterior Predictive Check

Key Findings

• Intercept: -2116.53 (95% CI: [-2131.41, -2100.91])

  • Indicates significantly shorter delays without weather incidents.

• Weather Count Coefficient: 1041.97 (95% CI: [1036.73, 1047.15])

  • A 1-unit increase in weather incidents leads to an average 1042-minute delay.

  • Weather incidents are infrequent but highly disruptive.

• Uncertainty Measures:

  • Residual variability: Standard deviation = 8676.19.

  • Suggests other unmeasured factors affecting delays.

• Model Diagnostics:

  • Rhat = 1.00 for all parameters, indicating convergence.

  • Large effective sample sizes ensure reliable posterior estimates.

Conclusion

• Key Insight:

  • Weather-related incidents, though infrequent, have a disproportionately large impact on delay times.

  • Highlights the need for better weather management and forecasting.

• Bayesian Approach:

  • Accounts for uncertainty, providing credible intervals for estimates.

  • Supports informed decision-making in airline operations and policy-making.

Discussion and Future Research

• What other factors could be included in the model?

• How could expanding the dataset improve insights?

• What advanced Bayesian methods could be explored?

• How should outliers be addressed?

• What assumptions should be revisited?

Thank You! Questions?